I understand the idea and math behind the concept of control variate for the sake of variance reduction, but I struggle to apply it to option pricing.
I need to simulate an European option of a stock which has the traditional charateristics:
- $B(S_t) = max(S_t -K,0)$
- The log-returns are normally distributed with parameters $(R-(\sigma^2/2))t$ and $\sigma_t$, with $R$ being the risk-free rate
I need to use a second European call option as control variate with strike $K_2$ and price $C_0$ which is assumed ot be close enough to the first option.
I need to give a algorithhm to simulate the price of the first option using the strike price and price.
I know that I need an high correlation to have an effective variance reduction and that the variance reduction can be computed as $Cov(X,Y)/Var(Y)$ but i don't see how to apply it in that context